Chowla–Mordell theorem

Chowla–Mordell theorem In mathematics, the Chowla–Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman Chowla and Louis Mordell, around 1951.

In detail, if {displaystyle p} is a prime number, {displaystyle chi } a nontrivial Dirichlet character modulo {displaystyle p} , and {displaystyle G(chi )=sum chi (a)zeta ^{a}} where {displaystyle zeta } is a primitive {displaystyle p} -th root of unity in the complex numbers, then {displaystyle {frac {G(chi )}{|G(chi )|}}} is a root of unity if and only if {displaystyle chi } is the quadratic residue symbol modulo {displaystyle p} . The 'if' part was known to Gauss: the contribution of Chowla and Mordell was the 'only if' direction. The ratio in the theorem occurs in the functional equation of L-functions.

References Gauss and Jacobi Sums by Bruce C. Berndt, Ronald J. Evans and Kenneth S. Williams, Wiley-Interscience, p. 53. Categories: Cyclotomic fieldsZeta and L-functionsTheorems in number theory